Calculation Method of the Distribution Statistics of Kolmogorov-Smirnov Type in Testings with Alternate Load for Finite Sample Size

In theory, the accelerated tests are implementing using widely the testings in an alternating mode. These testings are designed to determine the same scaling functions for recalculating of the accelerated tests results to the normal mode for any parts of single-type products. An improved method for testing in an alternating mode is proposed that allows estimating scaling functions for the recalculating with higher accuracy. The calculation method of the exact distributions statistics of Kolmogorov-Smirnov type designed to test hypotheses about the form of link function is given. The feature of these methods is the application of the Kaplan-Meier estimation of the reliability function to reduce the amount and duration of the alternative modes testings. Knowledge of the exact distributions of applied statistics is important due to the fact that in reality there are no large sample size of products allocated for testings. However, in most cases the asymptotic distributions of the statistics are used. It often leads to large errors in the analysis of test results. The proposed method can eliminate this drawback.

References

[1] Nelson W. Accelerated Testing Statistical Models, Test Plans and Data Analysis. New Jersey, John Wiley & Sons, 2004. 601 p.

[2] Kartashov G.D. Forsirovannye ispytaniya apparatury [Accelerated testing of equipment]. Moscow, Znanie Publ., 1986. 54 p.

[3] Wasserman L. All of Nonparametric Statistics. N.Y., Springer Science+Business Media, 2006. 272 p.

[4] Gamiz M.L., Kulasekera K.B., Limnios N., Lindqvist H. Applied Nonparametric Statistics in Reliability. London, Springer-Verlag, 2011. 227 p.

[5] Kartashov G.D. Predvaritel’nye issledovaniya v teorii forsirovannykh ispytaniy [Preliminary researches in the theory of accelerated testing]. Moscow, Znanie Publ., 1980. 51 p.

[6] Kartashov G.D. Establishing links between unobserved simultaneously random variables. Sb. nauch. tr. Instituta mat. i kib. AN LitSSR. "Primenenie teorii veroyatnostey i matematicheskoy statistiki" [Collect. Pap. Ins. Math. Informatics of Acad. Sci. LitSSR "Application Probability Theory and Mathematical Statistics"], 1981, no. 4, pp. 18-29 (in Russ.).

[7] Timonin V.I. The exact distribution of rank statistics for censored data in testing in with varying load. Sb. Tr. MGTU im. N.E. Baumana [Proc. Bauman Moscow Higher Technical School], 1985, no. 428, pp. 48-54 (in Russ.).

[8] Timonin V.I. Optimization of the preliminary studies in the theory of accelerated testings. Vestn. Mosk. Gos. Tekh. Univ. im. N.E. Baumana, Estestv. Nauki [Herald of the Bauman Moscow State Tech. Univ., Nat. Sci.], 2004, no. 1, pp. 22-33 (in Russ.).

[9] Timonin V.I., Ermolaeva M.A. Kaplan-Meier estimates in Kolmogorov-Smirnov statistics for hypothesis testing within testing in alternative modes. Elektromagn. volny i electron. sist. [Electromagn. Waves and Electron. Systems], 2010, vol. 15, no. 7, pp. 18-26 (in Russ.).

[10] Klein J.P., Moeschberger L. Survival analysis techniques for censored and truncated data. N.Y., Springer-Verlag, 2003. 536 p.

[11] Meeker W.Q., Escobar L.A. Statistical methods for reliability data. N.Y., Wiley, 1998. 701 p.

[12] Sidak H., Hajek J. Theory of rank tests, 2nd ed. Academic Press, 1999. 435 p. (Russ. Ed.: Gaek Ya., Shidak Z. Teoriya rangovykh kriteriev. Moscow, Nauka Publ., 1971. 376 p.).

[13] Elandt-Johnson R., Johnson N. Survival models and data analysis. N.Y., Wiley, 1980. 403 p.

[14] Sarhan, A.E., Greenberg B.G. (Ed.) Contributions to Order Statistics. John Wiley & Sons, New York-London, 1962, p. 482. (Russ. Ed.: Sarkhan A.E., Grinberg B.G., eds. Vvedenie v teoriyu poryadkovykh statistic. Moscow Statistika Publ., 1970. 343 p.).

[15] Sheshkin D.J. Handbook of parametric and nonparametric statistical procedures. Boca Raton, Chapman & Hall/CRC, 2000. 1002 p.